Forcing for independence proofs
Dr. Andrew Brooke-Taylor,
Taught Course Centre,
October - December 2010.
The handout given out in the final lecture had some typos;
this electronic copy has fewer.
Lecture notes and whiteboard images
Summary from before the course
Forcing is the key technique used by set theorists in proving independence
proofs, and its development by Paul Cohen won him the Fields Medal in
1966. In this course I will show how it can be used to prove that the
Continuum Hypothesis is independent of the standard ZFC axioms for set theory,
and to prove that the Axiom of Choice is independent of the other
No prior knowledge of set theory will be assumed, however a basic level of
mathematical logic (eg what a first order formula is) would be helpful.
I was hoping to also cover
- Introduction: what independence means (definition, not philosophy!),
models of set theory.
- Generic extensions, the forcing relation
- Definability of the forcing relation
- The Truth Lemma, extensions satisfy ZFC
- Chain conditions, Con(ZFC) implies Con(ZFC + ~CH)
- Closure conditions, Con(ZFC) implies Con(ZFC + CH)
but we didn't get to them.
- Symmetric models, Con(ZF) implies Con(ZF + ~AC)
- Looking ahead and at other techniques: ways to prove Con(ZF) implies
The main reference I will be working from is Kunen, Set theory
(subtitle: An introduction to indepence proofs), North-Holland, 1980,
chapter 7. This does not cover the AC part (which we actually didn't get to in the end).
For that, I suggest Jech, The Axiom of Choice, Dover, 2009.
Last updated: 17 December 2010.